We study cosmic structures in the quadratic Degenerate Higher Order Scalar Tensor (qDHOST) model, which has been proposed as the most general scalar-tensor theory (up to quadratic dependence on the covariant derivatives of the scalar field), which is not plagued by the presence of ghost instabilities. We then study a static, spherically symmetric object embedded in de Sitter space-time for the qDHOST model. This model exhibits breaking of the Vainshtein mechanism inside the cosmic structure and Schwarzschild-de Sitter space-time outside, where General Relativity (GR) can be recovered within the Vainshtein radius. We then look for the conditions on the parameters on the considered qDHOST scenario which ensure the validity of the Vainshtein screening mechanism inside the object and the fulfilment of the recent GW170817/GRB170817A constraint on the speed of propagation of gravitational waves. We find that these two constraints rule out the same set of parameters, corresponding to the Lagrangians that are quadratic in second-order derivatives of the scalar field, for the shift symmetric qDHOST. The authors of Ref. [8,9] claimed that their extension of the Horndeski theory to the so-called Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theory, leads to a new class of models without the Ostrogradsky ghost instability. A later proposal showed that higher-order derivatives in the Lagrangian may not necessarily introduce Ostrogradsky ghosts, provided certain degeneracy conditions are met [10]. Indeed, "degenerate" Lagrangians with non-invertible kinetic matrix will ensure that the number of degrees of freedom is preserved, thus making the theory free from the Ostrogradsky ghost; this new class was named "degenerate higher order scalartensor" (DHOST) theory [11][12][13]. A particular extension of Horndeski was proposed earlier in Ref. [14], which appeared as a result of a disformal transformation on the Einstein-Hilbert action, later found a specific subclass of HOST theory. DHOST theory is categorised into several classes [13]. Class I DHOST theories are the only one which are healthy from the gradient instability, i.e., the square of the speed of the tensor modes (gravitational-wave speed) and that of the scalar mode (sound speed) do not have opposite sign, c 2 s ∝ −c T 2 [15]. Gravity is well tested and established on small scales (e.g. laboratory, solar system, ...). Therefore, there must be a screening mechanism able to suppress the fifth-force mediated by the new scalar degree of freedom, without destroying the modifications on large scales, while recovering GR on a small scale. In general, the so-called Vainshtein screening is widely used for higher-order scalar-tensor theories [16]. In the Vainshtein screening, the non-linear selfinteractions of the scalar field suppress the propagation of the fifth-force near the matter source [17,18]. The Vainshtein mechanism in the Horndeski framework has been studied intensively

This paper investigates the propagation of cosmological gravitational waves interacting with free-streaming neutrinos within the context of Horndeski theories of gravity constrained by the detection of GW170817. We apply the theory of cosmological perturbations to explicitly derive the Einstein-Boltzmann equation for the damped propagation of firstorder transverse traceless gravitational waves. In contrast to general relativity, we argue that modified gravity can give rise to non-vanishing free-streaming damping effects during the cosmological matter dominated era. We also provide an analytic formula for the main multipole order with which modified gravity and free-streaming neutrinos damp the variety of tensor correlation functions of the cosmic microwave background.

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